Embedded newform 37.4.h.a.3.4

• Label: 37.4.h.a.3.4
• Level: $37$
• Weight: $4$
• Character: 37.3
• Analytic conductor: $2.183$
• Analytic rank: $0$
• Dimension: $48$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 37 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	37.h (of order \(18\), degree \(6\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(2.18307067021\)
Analytic rank:	\(0\)
Dimension:	\(48\)
Relative dimension:	\(8\) over \(\Q(\zeta_{18})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{18}]$

Embedding invariants

Embedding label			3.4
Character	\(\chi\)	\(=\)	37.3
Dual form			37.4.h.a.25.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.746486 + 0.889628i) q^{2} +(3.53224 - 2.96390i) q^{3} +(1.15499 + 6.55027i) q^{4} +(3.46112 - 9.50934i) q^{5} +5.35489i q^{6} +(25.4052 + 9.24674i) q^{7} +(-14.7354 - 8.50748i) q^{8} +(-0.996488 + 5.65136i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 48 q - 3 q^{2} + 9 q^{4} - 27 q^{5} - 108 q^{7} + 144 q^{8} + 42 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/37\mathbb{Z}\right)^\times\).

\(n\)	\(2\)
\(\chi(n)\)	\(e\left(\frac{13}{18}\right)\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

(See \(a_n\) instead)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.746486	+	0.889628i	−0.263923	+	0.314531i	−0.881689	−	0.471831i	\(-0.843593\pi\)
							0.617766	+	0.786362i	\(0.288038\pi\)
\(3\)	3.53224	−	2.96390i	0.679780	−	0.570403i	−0.236162	−	0.971714i	\(-0.575890\pi\)
							0.915942	+	0.401310i	\(0.131445\pi\)
\(5\)	3.46112	−	9.50934i	0.309572	−	0.850541i	−0.683168	−	0.730261i	\(-0.739398\pi\)
							0.992740	−	0.120280i	\(-0.0383793\pi\)
\(7\)	25.4052	+	9.24674i	1.37175	+	0.499277i	0.919668	−	0.392698i	\(-0.128458\pi\)
							0.452085	+	0.891975i	\(0.350680\pi\)
\(11\)	13.3158	−	23.0636i	0.364987	−	0.632176i	−0.623787	−	0.781594i	\(-0.714407\pi\)
							0.988774	+	0.149418i	\(0.0477401\pi\)
\(13\)	−46.5477	+	8.20762i	−0.993078	+	0.175106i	−0.646499	−	0.762914i	\(-0.723768\pi\)
							−0.346579	+	0.938021i	\(0.612657\pi\)
\(17\)	−73.6547	−	12.9873i	−1.05082	−	0.185287i	−0.378538	−	0.925586i	\(-0.623573\pi\)
							−0.672278	+	0.740298i	\(0.734684\pi\)
\(19\)	−73.1037	−	87.1216i	−0.882692	−	1.05195i	−0.998278	−	0.0586536i	\(-0.981319\pi\)
							0.115587	−	0.993297i	\(-0.463125\pi\)
\(23\)	−53.1321	+	30.6758i	−0.481688	+	0.278102i	−0.721119	−	0.692811i	\(-0.756372\pi\)
							0.239432	+	0.970913i	\(0.423039\pi\)
\(29\)	169.777	+	98.0211i	1.08713	+	0.627657i	0.932812	−	0.360363i	\(-0.117347\pi\)
							0.154322	+	0.988021i	\(0.450681\pi\)
\(31\)			31.0341i			0.179803i	0.995951	+	0.0899015i	\(0.0286552\pi\)
							−0.995951	+	0.0899015i	\(0.971345\pi\)
\(37\)	−179.293	−	136.041i	−0.796636	−	0.604459i
							\(41\)	−56.8110	−	322.191i	−0.216400	−	1.22726i	−0.878461	−	0.477814i	\(-0.841429\pi\)
0.662062	−	0.749449i	\(-0.269682\pi\)
\(43\)			376.457i			1.33510i	0.744566	+	0.667548i	\(0.232656\pi\)
							−0.744566	+	0.667548i	\(0.767344\pi\)
\(47\)	−116.127	−	201.137i	−0.360401	−	0.624232i	0.627626	−	0.778515i	\(-0.284027\pi\)
							−0.988027	+	0.154283i	\(0.950693\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	37.4.h.a.3.4	✓	48
37.5	odd	36		1369.4.a.j.1.30		48
37.25	even	18	inner	37.4.h.a.25.4	yes	48
37.32	odd	36		1369.4.a.j.1.19		48

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
37.4.h.a.3.4	✓	48	1.1	even	1	trivial
37.4.h.a.25.4	yes	48	37.25	even	18	inner
1369.4.a.j.1.19		48	37.32	odd	36
1369.4.a.j.1.30		48	37.5	odd	36